{ "artifact_id": "qft.artifact.holography-quantum-gravity.wormholes-gravitational-path-integrals-and-ensembles.jt-double-trumpet-gluing-map", "title": "Disconnected Disks and JT Double-Trumpet Gluing", "kind": "topology-and-gluing schematic", "source_revision": 1, "source_date": "2026-08-29", "schematic": true, "not_to_scale": true, "reader_question": "For the same two asymptotic JT boundary data, what distinguishes the disconnected and connected contributions, and where does the double-trumpet gluing measure come from?", "takeaway": "The connected cylinder is a distinct topology obtained by gluing two trumpet exteriors along a geodesic seam and integrating its length and relative twist; connectedness alone does not establish an ensemble interpretation.", "scope": { "theory": "pure Euclidean Jackiw-Teitelboim gravity", "geometry": "connected, orientable, genus-zero surfaces with two labeled asymptotic boundaries, compared with two disconnected disks", "ads2_radius": "L_2 = 1", "schwarzian_coupling": "C > 0 and common to both asymptotic boundaries", "matter_insertions": "none", "status": "fixed-topology Euclidean benchmark; schematic geometry rather than a quantitative metric plot" }, "symbols": [ { "symbol": "beta_i", "meaning": "fixed inverse temperature, equivalently renormalized length data, on asymptotic boundary B_i", "integrated": false }, { "symbol": "C", "meaning": "positive Schwarzian boundary coupling, with the same dimensions as beta_i", "integrated": false }, { "symbol": "b", "meaning": "dimensionless length of the matched finite closed geodesic seam", "domain": "0 < b < infinity", "integrated": true }, { "symbol": "theta", "meaning": "relative Fenchel-Nielsen twist around the matched seam", "identification": "theta is periodic with period b", "integrated": true }, { "symbol": "S_0", "meaning": "JT topological coupling multiplying the Euler characteristic" } ], "panels": [ { "id": "A", "title": "Same boundary contract, different topology", "fixed_boundary_data": [ "B_1(beta_1)", "B_2(beta_2)", "common C > 0" ], "branches": [ { "case": "disconnected", "topology": "D_1 disjoint union D_2", "connected": false, "components": 2, "total_euler_characteristic": 2, "euler_weight": "exp(2 S_0)", "amplitude_role": "product of two one-boundary disk contributions" }, { "case": "connected", "topology": "M_(0,2), the cylinder or double trumpet", "connected": true, "components": 1, "euler_characteristic": 0, "euler_weight": "1", "amplitude_role": "connected two-boundary fixed-topology coefficient" } ], "comparison": { "euler_factor_ratio_connected_to_disconnected": "exp(-2 S_0)", "qualification": "This is only the ratio of topological Euler factors in the S_0 to infinity limit at fixed beta_i/C. The full amplitude ratio also contains beta-dependent boundary coefficients and any admitted fluctuation factors." } }, { "id": "B", "title": "Cut, twist, and glue the connected cylinder", "objects": [ { "object": "uncut double trumpet", "boundaries": [ "asymptotic beta_1", "asymptotic beta_2" ], "distinguished_curve": "one matched closed geodesic seam of length b" }, { "object": "left trumpet", "amplitude": "Z_T(beta_1,b)", "boundaries": [ "asymptotic beta_1", "finite geodesic b" ] }, { "object": "right trumpet", "amplitude": "Z_T(beta_2,b)", "boundaries": [ "finite geodesic b", "asymptotic beta_2" ] } ], "relations": [ { "from": "uncut double trumpet", "to": "left and right trumpets", "operation": "cut at the matched seam" }, { "from": "left and right trumpets", "to": "uncut double trumpet", "operation": "identify equal-length seams, integrate one relative-twist period, and integrate b" } ], "twist_measure": "integral from 0 to infinity db times integral from 0 to b dtheta equals integral from 0 to infinity b db", "gluing_equation": "Zhat_(0,2)(beta_1,beta_2) = integral_0^infinity b db Z_T(beta_1,b) Z_T(beta_2,b)", "exception": "The cylinder is an integrated fixed-topology family, not one full JT saddle. It is exceptional: no ordinary stable Weil-Petersson core volume V_(0,2) is inserted." } ], "canonical_trumpet": { "equation": "Z_T(beta,b) = sqrt(C/(2 pi beta)) exp[-C b^2/(2 beta)]", "conditions": [ "C > 0", "beta > 0", "b >= 0" ], "integrated_result": "Zhat_(0,2)(beta_1,beta_2) = sqrt(beta_1 beta_2)/(2 pi (beta_1 + beta_2))", "normalization_check": "C cancels from the integrated cylinder coefficient" }, "admission_conditions": [ { "condition": "connected topology is included in the declared topology sum", "failure": "excluding it makes the connected term zero by definition" }, { "condition": "the declared integration cycle supplies a nonzero coefficient for the connected sector or relevant configuration family", "failure": "zero intersection leaves a formal geometry but no contribution" }, { "condition": "zero modes, physical negative modes, ghosts, and normalization are consistently treated", "failure": "an unresolved unpaired physical negative mode blocks interpreting the displayed positive integral as the amplitude" } ], "claim_ceiling": { "licensed": "a connected contribution to the declared Euclidean gravitational path integral at the stated fixed-topology order", "not_licensed": [ "a Lorentzian causal or traversable channel", "a convergent all-topology gravitational path integral", "an ensemble interpretation without an independently specified ensemble", "a violation of factorization in one exact fixed boundary theory" ] }, "accessibility": { "reading_order": [ "panel A shared boundary contract", "panel A disconnected branch", "panel A connected branch", "panel A Euler-factor comparison", "panel B uncut cylinder", "panel B cut and reverse gluing operations", "panel B two trumpet pieces and relative twist", "panel B measure and gluing equation", "panel B exceptional-cylinder and claim-boundary note" ], "non_color_encoding": "Disconnectedness is shown by two separate disk shapes; connectedness by one continuous cylinder; asymptotic boundaries use heavy solid strokes; matched seams use dashed strokes; cutting and gluing use distinct solid and dashed arrows; every object and relation is directly labeled.", "canvas": "explicit white background for light, dark, monochrome, and print contrast", "motion": "none" }, "independent_checks": [ "chi(D_1 disjoint union D_2) = 2 and chi(M_(0,2)) = 0, so the connected-to-disconnected Euler-factor ratio is exp(-2 S_0)", "one twist period theta in [0,b) gives integral dtheta db = b db", "multiplying the two canonical trumpet kernels and integrating b gives sqrt(beta_1 beta_2)/(2 pi (beta_1 + beta_2))", "the integrated result is symmetric under beta_1 exchange beta_2", "at beta_1 = beta_2 the integrated result is 1/(4 pi)", "the Schwarzian coupling C cancels from the integrated cylinder coefficient", "the stable-core formula is not applied to the exceptional pair (g,n) = (0,2)", "topology exclusion, zero contour intersection, and an unresolved unpaired negative mode are distinct failure modes", "no arrow or label represents Lorentzian propagation or ensemble averaging" ], "sources": [ { "citation": "Saad, Shenker, and Stanford, JT Gravity as a Matrix Integral (2019), section 3.4.1, equations 133-138", "url": "https://arxiv.org/abs/1903.11115", "role": "canonical trumpet normalization, double-trumpet gluing measure, and integrated cylinder coefficient" }, { "citation": "Mertens and Turiaci, Solvable Models of Quantum Black Holes: A Review on Jackiw-Teitelboim Gravity (2023), sections 4.2.1-4.2.2", "url": "https://doi.org/10.1007/s41114-023-00046-1", "role": "independent review of the exceptional disk and cylinder amplitudes and their normalization" } ] }